Fallacies of Reasoning · But in general, accuracies, relative changes, and percentages are...

Fallacies of ReasoningHow We Fool Ourselves

Wust, Valentin

December 18, 2019

Contents

1 Introduction 2

2 Base-Rate / Prosecutor’s Fallacy 2

2.1 HIV Self-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Facial Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Hormonal Birth Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 SIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 p-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Gambler’s / Hot Hand Fallacy 9

3.1 Monte Carlo 1913 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Would You Pay for Transparently Useless Advice? . . . . . . . . . . . . . 9

4 Hindsight Bias 12

4.1 Clinicopathologic Conferences . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Determinations of Negligence . . . . . . . . . . . . . . . . . . . . . . . . 14

1

2 BASE-RATE / PROSECUTOR’S FALLACY

1 Introduction

There is a lot to be said about how others can try to fool us, and misrepresent facts

to suit their narrative. However, what is equally interesting is how humans can fool

themselves. One of the ways of fooling oneself are common fallacies of reasoning, of

which I will present some in the following.

2 Base-Rate / Prosecutor’s Fallacy

In the presentation that was given just before mine, there was a statement concerning an

accuracy: ”Humans are 54% accurate in recognizing lies.” This immediately provoked a

question about the corresponding base-rate of lies with which this value was obtained.

Now, in this concrete example, one could argue that an accuracy of 54% is unimpres-

sive regardless of the base rate. Always guessing ”lie” will achieve an accuracy that is

equal to the base-rate of lies, while always guessing ”no lie” will achieve an accuracy

of one minus the base-rate of lies. Therefore, by choosing one of those strategies, it is

always possible to achieve an accuracy of at least 50% by guessing in a manner that

is completely uncorrelated to the way in which the lie is presented. One might further

assume that humans are not actively worse at spotting lies than random chance, which

would mean that the base-rate of lies should lie between 46% and 54%.

But in general, accuracies, relative changes, and percentages are meaningless unless

the corresponding base-rate is known. What is usually called the base-rate fallacy is

that humans tend to ignore base-rates when presented with relative information.

2.1 HIV Self-Test

One topic where the base-rate fallacy can be observed is the presentation of medical

tests, in this example an HIV self-test. It works by using blood from a finger-prick, and

the test result will either be positive or negative. The specificity, i.e. the conditional

probability of getting a positive result if you are truly HIV positive p(P|HIV), of those

test is practically one. This is of course intentional, since the case where an infected

person is not recognized by the test should be avoided. A high sensitivity usually comes

at the price of a lower specificity, i.e. the conditional probability to get a negative result

if you are in fact HIV negative p(N|¬HIV). However, for this test it is still about 99.8%,

which seems to be reasonably close to one on first glance. This information can also be

2


presented in a table, as is done in Table 1, where the missing values have been calculated

by using that all columns have to add to one.

HIV positive HIV negativetest positive p(P|HIV) = 1 p(P|¬HIV) = 0.002test negative p(N|HIV) = 0 p(N|¬HIV) = 0.998

Table 1: Probabilities for getting negative and positive results if you are HIV negativeor positive.

Let us now take a look at how one of those tests is promoted, the following examples

are taken from the web page and instructions of use of the INSTI HIV test:

• ”Accuracy greater than 99%.”

• ”Two dots means your test result is positive. You are probably HIV positive.”

• ”Specificity is calculated by dividing the number of INSTI negative test results

by the number of truly HIV negative persons that were tested. The higher the

specificity the better the test is at correctly identifying truly non-infected persons.

The INSTI HIV Self Test has a specificity of 99.8% in a performance evaluation

conducted by untrained lay users. This means a positive result will be correct 998

out of every 1000 tests.”

The first statement is true, since the accuracy is defined as the share of true results

among all results. As both HIV positive and HIV negative persons will get a true result

in at least 99.8% of all cases, a population with any prevalence of HIV will get almost

only true results. But is this actually the number we care about? Additionally, if the

prevalence of HIV is lower than 0.2%, even a ”test” that only returned negative results

regardless of actual infection status would achieve this accuracy.

The second statement is closer to what we would actually want to know, that is, how

high the probability of having HIV is if the test result is positive. We have already

established that negative results rule out HIV, so the only remaining question is how

accurate positive results are, not the overall accuracy of the test. The third example

tries to answer this question, but confuses the specificity with the probability that a

positive result will be a true positive. However, using Bayes’ theorem, we can calculate

this probability p(HIV|P):

3


p(HIV|P) =p(P|HIV) p(HIV)

p(P)

=p(P|HIV) p(HIV)

p(P|HIV) p(HIV) + p(P|¬HIV) p(¬HIV)

=

(1 +

p(P|¬HIV)

p(HIV)(1− p(HIV))

)−1

Here, we used that p(P|HIV) ≈ 1 and p(¬HIV) = 1− p(HIV). A plot of the resulting

function of the prevalence of HIV p(HIV) is shown in Fig. 1. We can now insert actual

prevalences into this function to see how accurate positive results of this test actually

are. The prevalence of HIV in Germany is about one in 1000, which means that positive

results are only 33% accurate. However, in South Africa about 20% of people are infected

with HIV. In this high-prevalence population, the accuracy of positive results is almost

perfect at 99%.

0.0001 0.001 0.01 0.1 10%

20%

40%

60%

80%

100%99%

33%

95%

7%

p(HIV)

p(H

IV|P

)

Figure 1: True positives for different prevalences. The two high values are for SouthAfrica, the two low values for Germany. The red ones use the total prevalenceof HIV, the black ones the prevalence of undiagnosed HIV among those thatdo not have a diagnosis of HIV.

In practice, however, those who are already diagnosed with HIV should be excluded

from this calculation, since they are not the target audience for an HIV test. Since in

both Germany and South Africa about 87% of all infected are already diagnosed, the

4


prevalence of undiagnosed HIV among the remaining population is about 1 in 7000 for

Germany, and about 4% for South Africa, which gives an accuracy of 7% and 95%,

respectively. So for the still relatively high prevalence of 4%, positive results usually

indicate an infection, whereas in Germany they are close to meaningless. The problem

with communicating those numbers is that this calculation crucially depends on the

prevalence in the target population. If this test is marketed for certain high-risk groups,

the results are quite reliable. For example, there are homosexual communities in Ger-

many where the prevalence of undiagnosed HIV is probably in the percentage range, and

for them, it is justified to claim that a positive result probably indicates an infection.

But for the larger population, this is certainly not true, as we have just seen.

http://www.rwi-essen.de/unstatistik/86/

https://www.instihivtest.com/en/

2.2 Facial Recognition

A similar example concerns facial recognition. A joint project of the German ministry

of the interior, the federal police and the Deutsche Bahn tested the feasibility of facial

recognition in train stations as a means of detecting certain targets, such as Islamists

who might be prone to terrorism. There are a host of other issues regarding how they

obtained their results, for example the testing conditions were far from realistic. But

even the results they obtained under those conditions would make for a rather useless

system, although they were communicated as a huge success. They used a list of target

persons, and measured the sensitivity and specificity with which they could be identified

among non-targeted persons. The results are displayed in Table 2.

target not targetalarm 0.8 0.001

no alarm 0.2 0.999

Table 2: Probabilities of raising an alarm for persons that were either targeted or not.

Again, one could argue that the overall accuracy of their test is at least 80%. However,

as before, the accuracy of positive results is what is more relevant in practice. We can

again calculate it using Bayes’ theorem, but first we need the base-rate of targets. This

technology was promoted as a way of alerting the police when potential Islamist terrorist

frequent train stations, so we will estimate the corresponding base-rate. In Germany,

about 12 million people take the train every day. On the other hand, there are only about

5


https://www.instihivtest.com/en/


600 known Islamists who are on what amounts to a terrorist watch-list. Assuming that

they take the train about as often as everybody else, about 100 of them will frequent a

train station on any given day, which amounts to a base-rate of only p(T) ≈ 8 × 10−6,

or one in 120,000. This means that the accuracy of alarms will only be p(A|T) = 1/125,

so for every alarm that indicates an Islamist, 124 false alarm will be raised. In practice,

this accuracy would be too small for any meaningful reaction to alarms, as you would

get 80 justified alarms every day, but also 12,000 false alarms. Again, the low base-rate

ruins everything.


2.3 Hormonal Birth Control

Another topic where the base-rate is usually ignored is hormonal birth control. Period-

ically, the high relative increase in thrombosis risk for women who use hormonal birth

control is mentioned, but the small absolute risk for young women is usually ignored.

Susan Solymoss. “Risk of venous thromboembolism with oral contraceptives”. In:

CMAJ : Canadian Medical Association journal 183.18 (2011)

2.4 SIDS

A fallacy that is closely related to the base-rate fallacy is the prosecutor’s fallacy, which

is again best illustrated with an example.

A British lawyer named Sally Clark had two children, in 1996 and 1998. Both of them

died from SIDS, i.e. sudden infant death syndrome. This is a diagnosis that is made if

an infant dies before its first birthday, and if no cause of death can be established, which

is not very common but does happen. A prosecutor then calculated the probability

of two children dying from SIDS to be one in 73 million and proceeded to indict her

for murder, arguing that this is so improbable that she must have killed her children.

She was subsequently convicted by a jury, with the sole evidence against her being this

probability, and spent three years in prison until she was eventually acquitted.

However, the prosecutor made two crucial mistakes. First, he assumed that the death

of her children were uncorrelated events, when in reality the chance of dying from SIDS

is much higher for children who already lost a sibling to it. A more reasonable estimate

6



of the probability for two sibling to both die of SIDS is one in 300,000, but still this

number in itself does not tell us anything. It expresses the conditional probability of

both her children dying, given that they died of SIDS p(both siblings died|SIDS), when

in reality, the probability that their children died from SIDS given that they are dead

p(SIDS|both siblings died) is the relevant quantity. We are not interested what the prior

probability for the death of her children is, since this event has already taken place. To

argue that she killed them, what is actually required is the chance that this was not due

to SIDS, and therefore due to murder, given that it already happened.

But in the example in subsection 2.1, we have seen that one cannot simply assume

that p(A|B) = p(B|A), since to get from one to the other we have to use Bayes’ theorem.

Crucially, the result will depend on the base-rates of the different conditions! So the

correct calculation for this example would compare the probability of her children dying

from SIDS with the probability of the alternative, i.e. that she killed them. As double

infanticide is comparatively unlikely, the result will be completely different from the

prior probability of either of them happening. One can try to estimate the probability

of her guilt, given that her children died, as was done in [Hil04], and it is probably

closer to 10%, a far cry from the one in 70 million chance implied by the prosecutor,

and certainly not large enough to convict her without any further evidence towards her

guilt. It is probably even smaller, since there are reasons to believe that the pathologist

actually overlooked evidence of an infection in her second child.

This is, however, a commonly used argument by prosecutors. Every time the chance

of a random agreement, e.g. for DNA tests, is mentioned, it is strongly implied to

correspond to the probability for the defendants innocence. In reality, one also needs to

consider the prior probability, since for example applying a DNA test to a large enough

population of innocent people will invariably produce false positives.

Intuitively, most people understand this. Nobody would automatically prosecute lot-

tery winners for fraud on the sole basis that winning the lottery is very unlikely. But as

with the base-rate fallacy, it is still easy to ignore the pitfalls of Bayesian statistics.

Ray Hill. “Multiple sudden infant deaths – coincidence or beyond coincidence?” In:

Paediatric and Perinatal Epidemiology 18.5 (2004)

2.5 p-Values

The prosecutor’s fallacy is also commonly applied to p-values. The p-value is defined

as the conditional probability of getting results that are as, or more, extreme as the

7


actual results, given that the null hypothesis it true, i.e. that the postulated effect

does not exist. However, we have just seen that it is, in principle, impossible to infer

the probability of the null hypothesis being false from just the p-value, as the prior

probability of the null hypothesis needs to be taken into account.

p(Data|Null) 6= p(Null|Data)

If the prior probability for an effect is very small, even a small p-value is not sufficient to

significantly improve the posterior probability, while hypotheses that are almost certainly

true should not be discarded even if a small p-value is not achieved. This is just a formal

version of saying that ”extraordinary claims require extraordinary evidence”.

This can also be used as an argument for why most published research findings might

be false. If one assumes that most tested hypotheses are unlikely to be true, and then

uses p < 0.05 to either confirm or discard them, one would expect to get many more

false positives than true positives.

One illustrative example concerns epidemiology. Consider an epidemiologist who has

randomly selected 50 participants for his study, and one of them happens to be an albino.

Now, being a good scientist, the researcher might calculate a p-value for this observation,

assuming that the test subjects are humans. Since albinism is quite rare, the probability

of getting one albino among 50 randomly selected humans is only about one in 400,

with gives a p-value of p ≈ 0.0025 < 0.05, which would, following the conventional logic,

be considered significant enough to reject the null hypothesis. But of course the null

hypothesis, i.e. that the subjects really are humans, is overwhelmingly likely, so this

p-value is not nearly small enough to lower the posterior probability by a large amount.

Jordan Ellenberg. How not to be wrong. London: Penguin Books, 2015, p. 136

8

3 GAMBLER’S / HOT HAND FALLACY

3 Gambler’s / Hot Hand Fallacy

3.1 Monte Carlo 1913

Figure 2: Roulette.

On August 18, 1913, roulette was played in the Casino in

Monte Carlo, when black came up fifteen times in a row. At

this point, gamblers, who thought that surely red was now

long overdue, began to bet large amounts of money on red.

However, the streak continued for another eleven rounds,

bringing the total to twenty-six times black. At this point,

the Casino had made some millions of francs.

The gamblers in his example exhibited the classic gam-

bler’s fallacy, they assumed that the past occurrence of black

or red influences their future likelihood. There are multiple

arguments for why this must be false. If this were indeed the case, the probability of

future events would depend on the amount of observations one takes into account. As

the roulette wheel should not care how many past results we observed, it cannot depend

on our observations. Also, probabilities are only guaranteed to hold in the limit of in-

finitely many observations. Streaks like the above are not actively compensated for, but

in this limit they are diluted until they no longer matter.

Darrell Huff and Irving Geis. How to take a chance. Gollancz, 1960, pp. 28-29

3.2 Would You Pay for Transparently Useless Advice?

Humans are prone to paying for objectively useless advice. Obvious examples for this

are fortune-tellers and astrologists, but also economic forecasts and investment advice.

For example, it has been shown that, on average, financial ”experts” at most barely

outperform the market, and usually not to a degree that justifies their high salaries.

However, humans tend to infer agency when presented with random occurrences.

This causes us to overestimate the influence that skill has on those predictions, and

underestimate the influence of chance. In particular, humans tend to attribute streaks

of good and bad performance to the skill of the predictor, when in reality they are to

a large degree due to chance. This is connected to the fact that humans tend to vastly

underestimate the occurrence of streaks in random data, since those simply do not ”feel”

random.

9


Powdhavee et al. investigated whether this effect also persists for obviously impos-

sible predictions in [PR15]. They used 378 undergraduate students from Thailand and

Singapore for their study, and asked them to bet on the outcome of coin flips. It was

made completely clear that the outcome of those coin flips was random. The coins were

provided by participants, flipped by randomly drawn participants, and both the coins

and the person who flipped them were repeatedly exchanged. In short, the outcomes

were clearly not predictable. The researchers then predicted them.

Before the experiment started, every participant was supplied with a random predic-

tion for all five coin tosses. They could pay money before every toss to open the relevant

prediction, if they did not do so they were still instructed to open them after every coin

toss. By random chance, some predictions were right, but of course the number of par-

ticipants who observed only correct predictions halved with every round. They observed

that those participants for whom all previous predictions were right were significantly

more likely to buy the next prediction (see Fig. 3), even though correct predictions were

obviously due to chance alone. This fallacy is usually called the hot-hand fallacy, where a

streak of correct predictions, bets etc. is assumed to be more likely than not to continue,

when in reality of course past random occurrences do not influence future probabilities.

In some sense, this is the opposite of the gambler’s fallacy.

Figure 3: Results of Powdhavee et al.

However, there are some points

of their analysis that can be crit-

icized. I observed that the confi-

dence intervals of their results ap-

proximately doubled between the

pre-print of the paper [PR12] and

the eventual (after three years)

published version, so I am prob-

ably not the first person to criti-

cize them. In general, those con-

fidence intervals are quite large

for the relevant groups. This

is of course caused by the real

randomness in their experiment,

since the number of people who

observe only correct predictions decays exponentially. They are still small enough to

support the general conclusion, but concrete values should be taken with a grain of salt.

10


0%

5%

8%10%

20%

30%

Per

cent

ofal

lfit

par

amet

ers

p < 0.1 p < 0.05 p < 0.01

Figure 4: Percent of parameters they indi-cated as significant below a cer-tain threshold, p < 0.1 presum-ably means 0.1 > p > 0.5 etc.

They do a rather elaborate statistical

analysis, which results in about 160 fit pa-

rameters, for each of which they indicated

whether it was significant with p < 0.1,

p < 0.05 and p < 0.01. I counted them,

Fig. 4 shows the percentage of parameters

in each p-value range.

The problem with using the normal

significance threshold here is that a lot

of false positives are expected by chance

alone, since they have so many parame-

ters that are tested for significance. This

is known as the problem of multiple com-

parisons. There are many ways of dealing

with this, the easiest version of course be-

ing to simply ignore it, as the authors did

here. A way of fixing the α error, i.e. the

probability of getting any false positives, is the Bonferroni correction. If one wants

an alpha error α, dividing it by the number of parameters m yields a new significance

threshold γ = α/m, using this results in the overall alpha error α. In this case, this

would result in the condition p < 0.0003 for significance, which most of the indicated pa-

rameters probably would not have crossed. However, this correction is quite drastic, and

therefore also strongly increases the β error, i.e. the probability of false negatives. Nev-

ertheless, it is usually used for studies that do huge numbers of comparisons, for example

in genome-wide association studies. A less drastic way is the Benjamini-Hochberg pro-

cedure, which uses the largest p value for which pk ≤ kmα is still true as the threshold. It

also controls the α error if one assumes that the comparisons are not independent, which

is probably better in an analysis like this, and here results in the condition p < 0.01.

But even with this, many of the parameters they indicated should not be counted as

significant.

Nattavudh Powdthavee and Yohanes E Riyanto. “Would you Pay for Transparently

Useless Advice? A Test of Boundaries of Beliefs in The Folly of Predictions”. In:

Review of Economics and Statistics 97.2 (2015)

11

4 HINDSIGHT BIAS

4 Hindsight Bias

4.1 Clinicopathologic Conferences

In the USA, it is traditional that hospitals hold clinicopathologic conferences. A presen-

ter, usually a young physician, is given all the relevant medical information for an old

case in which the patient ultimately died, except the final diagnosis. The presenter then

has to present the case to an audience, go through the possible diagnoses, and choose

the most likely one. Afterwards, the pathologist who did the actual autopsy announces

the correct diagnosis. As those cases are usually chosen for their difficulty, the presenter

frequently chooses an incorrect diagnosis.

Foresight Hindsight

26%

30%

34%

38%

42%

29.429.1

26.4

27.6

41.1 More Experienced

40.2 Less Experienced

38.5 Less Experienced

24.5 More Experienced

Harder

Easier

Pro

bab

ilit

yE

st.

ofC

orre

ctD

iagn

osis

Figure 5: Mean estimated probabilities of the correctdiagnosis by timing of the estimates (fore-sight vs. hindsight), experience of the es-timators (less, thin lines, vs. more, boldlines), and case difficulty (easier, solid lines,vs. more difficult, broken lines).

Dawson et al., [Daw+88], went

to four of those conferences, with

a total number of 160 attend-

ing physicians. They then di-

vided them into two groups, a

foresight (N=76) and a hindsight

(N=84) group. The foresight

group was asked to assign proba-

bilities to the possible diagnoses

after the presenter finished, the

hindsight group was given the

same task after the pathologist

had already stated the correct di-

agnosis. Two of those cases were

deemed harder than the other

two by expert physicians, and

for their analysis they further di-

vided the attendees into more

(N=75) and less (N=85) experienced physicians. Fig. 5 is the original graph of the

results from their paper. Three of the groups show a hindsight bias, but the group of

more experienced physicians for the harder cases actually shows a negative bias.

The difference in mean assigned probabilities for the actual diagnosis for the easier

cases was significant with p < 0.05. The same difference for the harder cases had

p = 0.06, ”which fell just short of the traditionally accepted significance level”. However,

due to the multiple comparisons problem, p < 0.05 is not actually the correct threshold.

12

4 HINDSIGHT BIAS

For the three groups that showed the hindsight bias, 30% ranked the correct diagnosis

first in foresight, and 50% in hindsight. Even though this analysis excludes one quarter

of their data, it is presented in the abstract without this caveat. They later say that 35%

of all foresight groups assigned the correct diagnosis the highest probability. It is not

given anywhere in the paper, but I would assume the corresponding number from the

hindsight group is probably closer to 45%, considering that one forth actually showed a

negative bias.

They did not provide any confidence intervals. They also did not mention the concrete

N for every group, it is probably about N=20 for each of them. Assuming a Poisson

distribution, which usually gives a reasonable estimate, one can calculate the error of

the mean as ∆ =√µ/N . Fig. 6a contains confidence intervals estimated in this way.

Foresight Hindsight

26%

30%

34%

38%

42%

Pro

bab

ilit

yE

st.

of

Cor

rect

Dia

gnosi

s

(a) With 95% Confidence Intervals.

Foresight Hindsight

26%

30%

34%

38%

42%

28

36

(b) Mean of all groups.

Figure 6: Improved versions of Fig. 5.

Due to the small size of each group, the confidence intervals are quite large. It is also

instructive to show the mean results for all groups, as in Fig. 6b, because in general, one

has to be very careful about subdividing the participants and analysing the subgroups

independently. This does not only result in the usual problem of multiple comparisons,

but one also strongly increases the number of comparisons beyond the number that is

actually done. Since the same arguments that the authors used to rationalize that one

subgroup did not show the bias could also be adapted to explain why any other group

did not show the bias, what is now relevant for judging the significance of the found

pattern is not simply its probability, but the probability of finding any similar pattern

13

4 HINDSIGHT BIAS

in the data. In this case, to judge the effect size they found when excluding one group,

one would have to calculate the probability of finding such an effect size when one is

allowed to exclude any similar group, e.g. the less experienced physicians etc., and when

the definitions of those groups are slightly altered. As this probability is usually rather

high, and as they did not specify the criteria by which they actually divided the groups,

one has to be very careful about analyses of this kind. In general, the mean results

for all participants are the relevant ones, but in this case they also show a significant

hindsight bias, i.e. the result seems more obvious in hindsight than in foresight.

Neal V. Dawson et al. “Hindsight Bias: An Impediment to Accurate Probability Esti-

mation in Clinicopathologic Conferences”. In: Medical Decision Making 8.4 (1988)

Hal R. Arkes. “The Consequences of the Hindsight Bias in Medical Decision Making”.

In: Current Directions in Psychological Science 22.5 (2013)

4.2 Determinations of Negligence

0%

20%

40%

60%

80%

100%

Was violence likely?

Was the therapist negligent?

Per

cent

ofP

arti

cip

ants

Agr

ee

Violent Not Specified Not Violent

Figure 7: Percentages of participants who affirmedthe above questions.

In [LL96], the researchers sent

out case studies to randomly

drawn people, of which about 300

replied. The case studies con-

tained one out of six different

cases, in all of which a thera-

pist had to decide how to han-

dle a potentially violent patient.

They were then told to imagine

themselves as jurors in a mal-

practice lawsuit, and were asked

whether they would convict the

therapist for criminal negligence.

Crucially, they sent out three dif-

ferent versions of the six cases.

For each of them, they varied

whether any outcome of the case was reported, and if, whether the patient ultimately

did or did not get violent and harmed another person. The participants were then asked

to answer eleven questions. The percentages of participants who agreed with two state-

ments, that violence was, a priori, likely, and that the therapist was criminally negligent,

14

4 HINDSIGHT BIAS

are shown in Fig. 7.

They observed that telling the participants that the patient got violent significantly

influenced their judgement of the case study, and that they were more likely to convict

the therapist for bad outcomes, irrespective of the actual reasonable behaviour.

0%

20%

40%

60%

80%

100%

Was violence likely?

Was the therapist negligent?P

erce

nt

ofP

arti

cip

ants

Agr

ee

Violent Not Specified Not Violent

Figure 8: Results from Fig. 7 with error bars (1σ).

Numbers for the different out-

come groups were not provided, I

had to calculate them myself to

be about 97, 94 and 102 for vi-

olent outcome, no outcome men-

tioned, and non-violent outcome,

respectively. They also did not

include any error bars in their

results. However, since they

should follow a Poisson distribu-

tion, they can be easily calculated

as ∆µ =√µ/N , and they seem

to be small enough (Fig. 8). The

problem of multiple comparisons

was again ignored, but this is not

so crucial here as they did a more modest number of significance tests.

Susan J LaBine and Gary LaBine. “Determinations of Negligence and the Hindsight

Bias”. In: Law and Human Behavior 20.5 (1996)

15

References

References

[Ark13] Hal R. Arkes. “The Consequences of the Hindsight Bias in Medical Decision

Making”. In: Current Directions in Psychological Science 22.5 (2013).

[Axe00] Stefan Axelsson. “The base-rate fallacy and the difficulty of intrusion detec-

tion”. In: ACM Transactions on Information and System Security (TISSEC)

3.3 (2000).

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